Stochastic analysis of 'simultaneous merge-sort'

The asymptotic behaviour of the recursion M-k =(d) Mk-1 boolean OR <(Mk-1)over bar> + Y-k is investigated; Y-k describes the number of comparisons which have to be carried out to merge two sorted subsequences of length 2(k-1) and M-k can be interpreted as the number of comparisons of 'Simultaneous Merge-Sort'. The challenging problem in the analysis of the above recursion lies in the fact that it contains a maximum as well as a sum. This demands different ideal properties for the metric in the contraction method. By use of the weighted Kolmogorov metric it is shown that an exponential normalization provides the recursion's convergence. Furthermore, one can show that any sequence of linear normalizations of M-k must converge towards a constant if it converges in distribution at all.